We research relations between optimal transport theory (OTT) and approximate Bayesian computation (ABC) possibly connected to relevant metrics defined on probability measures. Those of ABC are computational methods based on Bayesian statistics and applicable to a given generative model to estimate its a posteriori distribution in case the likelihood function is intractable. The idea is therefore to simulate sets of synthetic data from the model with respect to assigned parameters and, rather than comparing prospects of these data with the corresponding observed values as typically ABC requires, to employ just a distance between a chosen distribution associated to the synthetic data and another of the observed values. Our focus lies in theoretical and methodological aspects, although there would exist a remarkable part of algorithmic implementation, and more precisely issues regarding mathematical foundation and asymptotic properties are carefully analysed, inspired by an in-depth study of what is then our main bibliographic reference, that is Bernton et al. (2019), carrying out what follows: a rigorous formulation of the set-up for the ABC rejection algorithm, also to regain a transparent and general result of convergence as the ABC threshold goes to zero whereas the number n of samples from the prior stays fixed; general technical proposals about distances leaning on OTT; weak assumptions which lead to lower bounds for small values of threshold and as n goes to infinity, ultimately showing a reasonable possibility of lack of concentration which is contrary to what is proposed in Bernton et al. (2019) itself.
翻译:我们研究最佳运输理论(OTT)和近似巴伊西亚计算(ABC)之间的关系,它们可能与根据概率计量确定的相关指标相联系。ABC是基于巴伊西亚统计的计算方法,适用于特定基因模型,以便在可能性功能难以确定的情况下,估计其事后分布;因此,我们的想法是模拟模型中与指定参数有关的成套合成数据,而不是将这些数据的前景与ABC通常要求的相应观测值相比较,仅仅使用与合成数据相关的选定分布与观察到的值的另一种值之间的距离。我们的重点是理论和方法方面,尽管在算法执行中将存在显著的部分,并且根据对当时我们主要文献参考的Bernton等人(2019年)的深入研究,对数学基础和无症状特性的更精确问题进行仔细分析。我们的想法是,从模型中模拟出一套与ABC拒绝算法的严格拟订,同时从一个透明和一般的趋同结果,因为ABC阈值上升到零,而从先前的标本数中将有一个显著的部分,而从数学基础和无症状特性特性特性特性的更精确性的问题,最终将提出一个较弱的技术建议。